The compatibilities for \(f^\ast \) , \(\otimes \) , and \(j_!\)  for open immersions \(j\)  are easy and left to the reader. The compatibility for \(Rf_\ast \)  is a special case of Proposition REF  (1). For proper \(f\) , the claim for \(Rf_{\ast }=Rf_!\)  is [1]} This implies the claim for \(Rf_!\)  for compactifiable \(f\) . The result for \(Ri^!\)  in the case of a closed immersion follows from the triangle \(Ri^! \rightarrow i^{\ast } \rightarrow i^{\ast } Rj_\ast j^\ast \rightarrow \)  and its analytic counterpart, where \(j\)  is the complementary open immersion, using the known compatibilities for \(i^{\ast }\) , \(j^{\ast }\) , and \(Rj_{\ast }\) . Given these compatibilities, the claim for \(\mathop {R{H}\mathrm {om}}\)  now follows by induction on the dimension, imitating the devissage carried out in Corollary REF . The result for \(Ri^!\)  for general finite maps \(i: \mathcal {X} \rightarrow \mathcal {Y}\)  can be proven by following the argument in Corollary REF .
Finally, the claim for \(Rf^!\)  in the case \((n,p)=1\)  is local on the source, so we may factor \(f\)  as \(g\circ i\)  where \(g\)  is smooth of some pure relative dimension \(d\)  and \(i\)  is a closed immersion. Then \(Rf^! = Ri^! \circ Rg^! =Ri^! \circ g^{\ast }[2d](d)\)  by Poincaré duality for schemes, and \(Rf^{an !} = Ri^{an !} \circ g^{an \ast }[2d](d)\)  by Poincaré duality for rigid spaces as in Huber's book. The result now follows from the known compatibilities for \(g^{\ast }\)  and \(Ri^!\) .
